Smooth Initial Data Research Articles

Continuous solutions for a two-dimensional cross-diffusion problem involving doubly degenerate diffusion and logistic proliferation

Emerging from the modeling of intricate structures formed by bacterial motion in nutrient-deficient environments, the doubly degenerate nutrient taxis system [Formula: see text] is considered in smoothly bounded domains [Formula: see text]. In contrast to frameworks devoid of logistic growth where only small-data global solutions have so far been found in the literature, quadratic degradation in the logistic growth is shown to markedly amplify the overall dissipative effects to the extent that, for arbitrary smooth initial data, an associated no-flux type initial-boundary problem possesses global continuous weak solutions. Central to our findings is the identification of an advantageous dissipative structure subtly embedded in the system, the exploitation of which leads to the establishment of a positive lower bound for nutrients levels within any finite time interval, thereby ruling out the appearance of cross-degeneracies within such timeframes.

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

Tempered nonlocal integrodifferential equations with nonsmooth solution data

This work on the time discretization for the solution of the tempered nonlocal integro-differential equation with smooth and nonsmooth initial data are extended. We find the difficulty arising from theoretical analysis can be overcome by Laplace transform technique. A mount of proof skills have been provided based on Laplace transform technique for this kind of equations. The Laplace transform method is employed effectively to show that the proposed interpolating quadrature scheme is convergence for smooth and non-smooth initial data in both homogeneous and inhomogeneous cases.

Horizon phase spaces in general relativity

We derive a prescription for the phase space of general relativity on two intersecting null surfaces using the null initial value formulation. The phase space allows generic smooth initial data, and the corresponding boundary symmetry group is the semidirect product of the group of arbitrary diffeomorphisms of each null boundary which coincide at the corner, with a group of reparameterizations of the null generators. The phase space can be consistently extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces. The extended phase space includes the relative boost angle between the null surfaces as part of the initial data.We then apply the Wald-Zoupas framework to compute gravitational charges and fluxes associated with the boundary symmetries. The non-uniqueness in the charges can be reduced to two free parameters by imposing covariance and invariance under rescalings of the null normals. We show that the Wald-Zoupas stationarity criterion cannot be used to eliminate the non-uniqueness. The different choices of parameters correspond to different choices of polarization on the phase space. We also derive the symmetry groups and charges for two subspaces of the phase space, the first obtained by fixing the direction of the normal vectors, and the second by fixing the direction and normalization of the normal vectors. The second symmetry group consists of Carrollian diffeomorphisms on the two boundaries.Finally we specialize to future event horizons by imposing the condition that the area element be non-decreasing and become constant at late times. For perturbations about stationary backgrounds we determine the independent dynamical degrees of freedom by solving the constraint equations along the horizons. We mod out by the degeneracy directions of the presymplectic form, and apply a similar procedure for weak non-degeneracies, to obtain the horizon edge modes and the Poisson structure. We show that the area operator of the black hole generates a shift in the relative boost angle under the Poisson bracket.

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter α∈(0,1). This highlights the distinct feature of the alignment operator.

Polynomial Solutions of Electric Field Equations in Anisotropic Media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in electrically and magnetically anisotropic homogeneous media is considered in the paper. A method for the computation of the polynomial solutions of the initial value problem for the considered system is proposed. Symbolic computations are used and these symbolic computations are implemented in Maple. It is proved also that these polynomial solutions are approximate solutions of the considered initial value problem with smooth initial data and the inhomogeneous term. The computational experiments confirm the robustness of the suggested method for the computation of electric fields in general electrically and magnetically anisotropic media.

Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m>\max \{1,\,\frac {3N-2}{2N+2}\}$ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$ in an appropriate sense as $t\rightarrow \infty$ , where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$ . This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m>\frac {3N-2}{2N}$ in the case $N\geq 3$ , but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B22 (2017), 669–686) to $m>\frac {3N}{2N+2}$ when $N\geq 2$ .

Boundary and distributed optimal control for a population dynamics PDE model with discontinuous in time Galerkin FEM schemes

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.

On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data

Abstract We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems.

Unconditional H2$$ {H}^2 $$‐stability of the Euler implicit/explicit SAV‐based scheme for the 2D Navier–Stokes equations with smooth or nonsmooth initial data

AbstractIn this article, we propose ‐unconditional stable schemes for solving time‐dependent incompressible Navier–Stokes equations with smooth or nonsmooth initial data, , . The ‐stability analysis is established by leveraging the scalar auxiliary variable (SAV) approach. When dealing with nonsmooth initial data, we utilize a limited number of iteration of the semi‐implicit scheme followed by the SAV scheme. The overall efficiency is greatly enhanced due to the minimal computational cost of the semi‐implicit scheme and the explicit treatment of the nonlinear term within the SAV approach. The proposed schemes investigate two types of scalar auxiliary variables: the energy‐based variable and the exponential‐based variable. Rigorous proofs of the ‐unconditional stability of both schemes have been provided. Notice that both proposed numerical schemes enjoy unconditional long time stability for smooth and nonsmooth initial data when . Numerical experiments have been conducted to demonstrate the theoretical results.

The parametric Willmore flow

Abstract We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface). The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak Lipschitz W 2 , 2 W^{2,2} immersions. This fact opens in particular the possibility for defining the Willmore flow for weak Lipschitz W 2 , 2 W^{2,2} initial data.

Boundedness in a chemotaxis-May–Nowak model with exposed state

This paper is concerned with the Neumann initial–boundary value problem of the chemotaxis-May–Nowak model with exposed state for virus dynamics. It is proved that the problem admits a unique global classical solution which is uniformly bounded for all sufficiently smooth initial data in smoothly bounded domains Ω⊂Rn, n≤3.

Ill-posedness for the Half wave Schrödinger equation

We study the Cauchy problem for the half wave Schrödinger equation introduced by Xu [9]. There are some well-posedness results for the equation, however there is no ill-posedness result. We focus on the scale critical space and obtain the ill-posedness in the super-critical or at the critical space under certain condition. The proofs in the super-critical space are based on the argument established by Christ, Colliander and Tao [4]. More precisely, we analyze dispersionless equation with smooth initial data, namely the Schwartz function and it is locally well-posed in some weighted Sobolev space. We construct the solution for the half wave Schrödinger equation by using the solution for the dispersionless equation and we can exploit the norm inflation or the decoherence properties. For the critical space, we use the standing wave solution, which was proved the existence by Bahri, Ibrahim and Kikuchi [1].

High-order splitting finite element methods for the subdiffusion equation with limited smoothing property

In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in R2. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the W1,q-energy estimate (q>2) motivated by the partially inviscid two-dimensional Boussinesq system. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.

An approximation method for exact controls of vibrating systems with numerical viscosity

We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell’s “stabilizability implies controllability” principle and a finite elements method of order θ with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order θ for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.

On the regularity of the De Gregorio model for the 3D Euler equations

We study the regularity of the De Gregorio (DG) model \omega_{t} + u\omega_{x} = u_{x} \omega on S^{1} for initial data \omega_{0} with period \pi and in class X : \omega_{0} is odd and \omega_{0} \leq 0 (or \omega_{0} \geq 0 ) on [0,\pi/2] . These sign and symmetry properties are the same as those of the smooth initial data that lead to singularity formation of the De Gregorio model on \mathbb{R} or the generalized Constantin–Lax–Majda (gCLM) model on \mathbb{R} or S^{1} with a positive parameter. Thus, to establish global regularity of the DG model for general smooth initial data, which is a conjecture on the DG model, an important step is to rule out potential finite time blowup from smooth initial data in X . We accomplish this by establishing a one-point blowup criterion and proving global well-posedness for initial data \omega_{0} \in H^{1} \cap X with \omega_{0}(x) x^{-1}\in L^{\inf} . On the other hand, for any \alpha \in (0,1) , we construct a finite time blowup solution from a class of initial data with \omega_{0} \in C^{\alpha}\cap C^{\infty}(S^{1} \setminus \{0\}) \cap X . Our results imply that singularities developed in the DG model and the gCLM model on S^{1} can be prevented by stronger advection.

Global solutions of quasi-linear Hamiltonian mKdV equation

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space–time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.

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